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Theorem 2omotap 7260
Description: If there is at most one tight apartness on 2o, excluded middle follows. Based on online discussions by Tom de Jong, Andrew W Swan, and Martin Escardo. (Contributed by Jim Kingdon, 6-Feb-2025.)
Assertion
Ref Expression
2omotap (∃*𝑟 𝑟 TAp 2oEXMID)

Proof of Theorem 2omotap
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 2omotaplemst 7259 . . . . 5 ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝑥 = {∅}) → 𝑥 = {∅})
21ex 115 . . . 4 (∃*𝑟 𝑟 TAp 2o → (¬ ¬ 𝑥 = {∅} → 𝑥 = {∅}))
3 df-stab 831 . . . 4 (STAB 𝑥 = {∅} ↔ (¬ ¬ 𝑥 = {∅} → 𝑥 = {∅}))
42, 3sylibr 134 . . 3 (∃*𝑟 𝑟 TAp 2oSTAB 𝑥 = {∅})
54adantr 276 . 2 ((∃*𝑟 𝑟 TAp 2o𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})
65exmid1stab 4210 1 (∃*𝑟 𝑟 TAp 2oEXMID)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  STAB wstab 830   = wceq 1353  ∃*wmo 2027  wss 3131  c0 3424  {csn 3594  EXMIDwem 4196  2oc2o 6413   TAp wtap 7250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-tr 4104  df-exmid 4197  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-1o 6419  df-2o 6420  df-pap 7249  df-tap 7251
This theorem is referenced by:  exmidmotap  7262
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